Fundamentals of Probability and Statistics for Engineers

The corresponding probability distribution function is

where m is the largest integer less than or equal to x. The mean and variance of
X can be found as follows:

In the above, the interchange of summation and differentiation is allowed
because 1. Following the same procedure, the variance has the form

Example 6.5.Problem: a driver is eagerly eyeing a precious parking space
some distance down the street. There are five cars in front of the driver, each of
which having a probability 0.2 of taking the space. What is the probability that
the car immediately ahead will enter the parking space?
Answer: for this problem, we have a geometric distribution and need to
evaluate for and Thus,

...

pX(k)

p

qp

q^2 p

k

1234567

Figure6. 1 Geometricdistribution

168 FundamentalsofProbabilityandStatisticsforEngineers

FX…x†ˆ

Xmx

kˆ 1

pX…k†ˆp‡qp‡‡qm^1 p

ˆ… 1 q†… 1 ‡q‡q^2 ‡‡qm^1 †ˆ 1 qm; … 6 : 15 †

EfXgˆ

X^1

kˆ 1

kqk^1 pˆp

X^1

kˆ 1

d

dq

qk

ˆp

d

dq

X^1

kˆ 1

qkˆp

d

dq

q

1 q



ˆ

1

p

: … 6 : 16 †

jqj<

^2 Xˆ

1 p

p^2

: … 6 : 17 †

pX(k) kˆ 5 pˆ 0 :2.

pX… 5 †ˆ… 0 : 8 †^4 … 0 : 2 †ˆ 0 : 82 ;

pX(k)